EXTENSIONAL REALIZABILITY AND CHOICE FOR DEPENDENT TYPES IN INTUITIONISTIC SET THEORY

In [17], we introduced an extensional variant of generic realizability [22], where realizers act extensionally on realizers, and showed that this form of realizability provides inner models of CZF (constructive Zermelo-Fraenkel set theory) and IZF (intuitionistic Zermelo-Fraenkel set theory), that further validate AC(FT )(the axiom of choice in all finite types). In this paper, we show that extensional generic realizability validates several choice principles for dependent types, all exceeding AC(FT). We then show that adding such choice principles does not change the arithmetic part of either CZF or IZF.